Sergey G. Bobkov
For noncorrelated random variables, we study a concentration property of the family of distributions of normalized sums formed by sequences of times of a given large length.
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We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular we show that previously known result from Rempa\la and Weso\lowski (Statist. Probab. Lett. 74 (2005) 129--138), which can be obtained by applying the logarithm as the function, holds true under weaker assumptions.
We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry-Esséen bound with respect to several distances.
In this paper, we prove a Berry--Esseen bound with optimal order for self-normalized sums of local dependent random variables under some mild dependence conditions. The proof is based on Stein's method and a randomized concentration inequality. As applications, we obtain optimal Berry--Esseen bounds for $m$-dependent random variables and graph dependency.
We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} : i\geq 1} $ are i.i.d. random variables, independent of ${Y_{i} : i \geq 1}$. The main result of the paper is that each subsequential limit law of T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.
Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ ψ_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student's $t$-statistic or the empirical correlation coefficient.
Moments for self-normalized partial sums
本文研究尾部指数小于2的正则变化平稳序列的自正则化部分和相关比值统计量的矩存在性，发现其矩存在性与序列的极值簇相关。
Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the self-normalized partial sums processes $\{S_{[nt]}/V_n, 0\le t\le 1\}$ when $X$ belongs to the domain of attraction of a stable law with index $α\in (0,2]$. The respective limiting distributions of the random variables ${\max_{1\le i\le n}|X_i|}/{S_n}$ and ${\max_{1\le i\le n}|X_i|}/{V_n}$ are also obtained under the same condition.
On large deviation probabilities for self-normalized sums of random variables
本文通过将自归一化随机游走的大偏差问题转化为二元随机游走问题，在更一般条件下证明了Shao的经典定理，并给出了精确的大偏差渐近及图形解释。
We consider an Ornstein-Uhleneck (OU) process associated to self-normalised sums in i.i.d. symmetric random variables from the domain of attraction of $N(0, 1)$ distribution. We proved the self-normalised sums converge to the OU process (in $C[0, \infty)$). Importance of this is that the OU process is a stationary process as opposed to the Brownian motion, which is a non-stationary distribution (see for example, the invariance principle proved by Csorgo et al (2003, Ann Probab) for self-normalised sums that converges to Brownian motion). The proof uses recursive equations similar to those that…
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.
A well-known longstanding conjecture on the supremum of the tails of normalized sums of independent Rademacher random variables is disproved. A related conjecture, also recently disproved, is discussed.
We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the Rényi entropy and Tsallis entropy of order $α (α>0)$ of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding Rényi entropy and Tsallis entropy of the Gaussian limit, and obtain sharp rates of convergence.
Self-normalized Sums in Free Probability Theory
该研究在自由概率论框架下证明自由自伴随机变量的自归一化和弱收敛于Wigner半圆律，给出了有界和无界情形下的Kolmogorov距离收敛速度，建立了经典自归一化极限定理的自由版本。
We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail-accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized …
We investigate the invariance principle for set-indexed partial sums of a stationary field $(X\_{k})\_{k\in\mathbb{Z}^{d}}$ of martingale-difference or independent random variables under standard-normalization or self-normalization respectively.
Let X_{n} be an integer valued Markov Chain with finite state space. Let S_{n}=\sum_{k=0}^{n}X_{k} and let L_{n}(x) be the number of times S_{k} hits x up to step n. Define the normalized local time process t_{n}(x) by t_{n}(x)=\frac{L_{n}(\sqrt{n}(x)}{\sqrt{n}}. The subject of this paper is to prove a functional, weak invariance principle for the normalized sequence t_{n}, i.e. we prove that under some assumptions about the Markov Chain, the normalized local times converge in distribution to the local time of the Brownian Motion.
In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(α)$, has a non trivial distribution iff $p=α=2$. The case for $2 > p > α$ and $p \le α< 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörgö et al. who showed Donsker's …
We study the joint limit behavior of sums, maxima and $\ell^p$-type moduli for samples taken from an $\mathbb{R}^d$-valued regularly varying stationary sequence with infinite variance. As a consequence, we can determine the distributional limits for ratios of sums and maxima, studentized sums, and other self-normalized quantities in terms of hybrid characteristic functions and Laplace transforms. These transforms enable one to calculate moments of the limits and to characterize the differences between the iid and stationary cases in terms of indices which describe effects of extremal clusterin…
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of random variables.
We study p-adic counterparts of stable distributions, that is limit distributions for sequences of normalized sums of independent identically distributed p-adic-valued random variables. In contrast to the classical case, non-degenerate limit distributions can be obtained only under certain assumptions on the asymptotic behaviour of the number of summands in the approximating sums. This asymptotics determines the ``exponent of stability''.
Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér type moderate deviation theorem for the general self-normalized sum \sum_{i=1}^n X_i/(\sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramér (1938) theorem and the self-normalized Cramér type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Speci…
We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their associated distributions -- the binomial distributions for the binomial coefficients and the Irwin-Hall distributions (uniform B-splines) for the Eulerian numbers.
The Berry-Esseen Bound for High-dimensional Self-normalized Sums
该研究针对高维自归一化统计量逐坐标最大值的高斯近似问题，在弱矩假设下推导出显式Berry-Esseen界，填补了高维CLT领域的相关研究空白。
This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.
We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space $D[0,1]$. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error type processes.
We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums
本文针对$2<p\le3$的有限$p$阶绝对矩条件，建立部分和自正则化最大值的精确Cramér型中偏差界，证明Shao2010的猜想，通过反射游走表示等方法完成证明。
We derive in this article the asymptotic behavior as well as non-asymptotical estimates of tail of distribution for self-normalized sums of random variables (r.v.) under natural classical norming. We investigate also the case of non-standard random norming function and the tail asymptotic for the maximum distribution for self-normalized statistics. We do not suppose the independence or identical distributionness of considered random variables, but we assume the existence and sufficient smoothness of its density. We show also the exactness of our conditions imposed on the considered random vari…
A functional limit theorem for self-normalized linear processes with random coefficients and i.i.d. heavy-tailed innovations
本文研究了具有随机系数和i.i.d.厚尾创新的严格平稳线性过程的自归一化功能极限定理，证明了在特定条件下部分和有界时的收敛性。
An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.
Convergence in $χ^2$ Distance to the Normal Distribution for Sums of Independent Random Variables
本文证明在独立同方差零均值随机变量的 $\chi^2$ 距离平均值有界时，其标准化和的 $\chi^2$ 距离以 $O(1/n)$ 速率收敛于正态分布。
We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ together with a sequence of independent, identically distributed $X$-space valued random variables $ξ_1,\dots,ξ_n$ and give a good estimate on the tail distribution of $\sup_{f\in\Cal F}\sum_{j=1}^n f(ξ_j)$ if the expected values $E|f(ξ_1)|$ are very small for all $f\in\Cal F$. In a subsequent paper~[2] we shall give a sharp bound for the supremum of normalized sums of i.i.d. random variables in a more general case. But that estimate is a consequence of the results in this work.
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
Asymptotic results for sums and extremes
本文研究了非中心中偏差问题，证明了有上界独立同分布随机变量的和与最大值二元序列、无上界随机变量以及指数随机变量部分最小值之和的中偏差结果。
In this paper, we study almost sure central limit theorems for multiple stochastic integrals and provide a criterion based on the kernel of these multiple integrals. We apply our result to normalized partial sums of Hermite polynomials of increments of fractional Brownian motion. We obtain almost sure central limit theorems for these normalized sums when they converge in law to a normal distribution.
The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(Ω, \mathscr{H}, \widehat{\mathbb E})$. Denote $S_n=\sum_{k=1}^n X_k$ and $V_n^2=\sum_{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n \ge x_n \}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables…
It is proved that infinitesimal triangular arrays obtained from normalized partial sums of strongly mixing (but not necessarily stationary) random sequences, can produce as lilmits only selfdecomposable distributions.
Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$
研究在单位正方形内由泊松点过程生成的点集，基于德劳内三角剖分考虑归一化增量和，推导当点密度趋于无穷时的中心极限定理。
Let $\{V_{i,j}; (i,j)\in\N^2\}$ be a two-dimensional array of i.i.d.\ random variables. The limit laws of the sum of independent random products $$ Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n} e^{V_{i,j}} $$ as $n,N_n\to\infty$ have been investigated by a number of authors. Depending on the growth rate of $N_n$, the random variable $Z_n$ obeys a central limit theorem, or has limiting $α$-stable distribution. The latter result is true for non-lattice $V_{i,j}$ only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence $Z_n$ fails to converge in distribution, i…
We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller [8] as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev [10, 11]. As an application of our result, motivated by…
In this note, we proved that weak limits, of sums of independent positive identically distributed random variables which are re-normalized by a non-linear shrinking transform $\max(0, x-r)$, are either degenerate or (some) compound Poisson distributions.
Consider a random $d$-dimensional simplex whose vertices are $d+1$ random points sampled independently and uniformly from the unit sphere in $\mathbb R^d$. We show that the expected sum of solid angles at the vertices of this random simplex equals $\frac 18$ if $d=3$ and $\frac{539}{288π^2}-\frac 16$ if $d=4$. The angles are measured as proportions of the full solid angle which is normalized to be $1$. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected angle-sums of random…
For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the possible limit distributions are precisely the operator-selfdecomposable laws.
In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences $p^{it}$, $χ(p)$, $χ_d(p)$, $λ_f(p)$, and $\text{Kl}_q(a-n, b)$; where $p$ ranges over the primes, $t$ varies in a large interval, $χ$ varies among all characters modulo $q$, $χ_d$ varies over quadratic characters attached to fundamental discriminants $|d|\leq x$, $λ_f(n)$ are the Fourier coefficients of holomorphic cus…
Refined Berry-Esseen bounds under local dependence
本文在局部依赖下通过Stein方法和新集中不等式，改进了自归一化和非自归一化和的Berry-Esseen界，并应用于图依赖等场景。
Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums
本文研究了随机矩阵对角和分布的光滑性与Lévy集中函数，通过引入广义归一化矩阵hafnian的新辅助不等式，给出了全变差距离和集中函数的新显式上界。
We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem.
Weak dependence and optimal quantitative self-normalized central limit theorems
本文针对弱相依平稳序列，证明了自归一化部分和的Berry-Esseen界达到最优阶$n^{-1/2}$，并指出最优长程方差估计器导致次优界，通过特定范围内的过度平滑可恢复最优收敛速度，适用于多种动力系统与时间序列模型。
Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a Markov chain {Phi(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained: 1. Spectral theory: Well-behaved solutions can be constructed for the ``multiplicative Poisson equation''. 2. A ``multiplicative'' mean ergodic theorem: For all complex αin a neighborhood of the origin, the normalized mean of \exp(αS_t) converges exponentially fast to a solution of the multiplicative Poisson equation. 3. Edg…
By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$ρ(F_n,Φ)\le\frac{0.335789(β^3+0.425)}{\sqrt{n}}$$ and $$ρ(F_n,Φ)\le \frac{0.3051(β^3+1)}{\sqrt{n}} $$ are proved for the uniform distance $ρ(F_n,Φ)$ between the standard normal distribution function $Φ$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $β^3$. The first of these inequalities sharpens the best known version of the clas…
We show the existence of a limiting distribution $\cD_\cC$ of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set $\cC$. Using a correspondence between the small divisors in the Fourier series of the discrepancy function and lattices with short vectors, and mixing of diagonal flows on the space of lattices, we identify $\cD_\cC$ with the distribution of the level sets of a function defined on the product of the space of lattices with an infinite dimensional torus. We apply our results to counting lattice points in slanted cylinder…
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
研究两个独立同分布各向同性分数布朗运动场在固定二维域上的点wise最大值的平方增量和，揭示其在Hurst参数H<1/2时的渐近行为，证明归一化后收敛于确定常数乘以两个场差的局部时间。
We derive exponential bounds for tail of distribution for natural, i.e. under ordinary logarithm, normalized sums of arrays of random variables, not necessarily independent.
Esscher Transform and the Central Limit Theorem
本文研究高维欧氏空间中的Esscher变换在中心极限定理中的应用，利用无穷阶Rényi散度探索了i.i.d.随机向量标准化和的正态逼近充分必要条件，推广了近期的一维结果。
We study the free central limit theorem for not necessarily identically distributed free random variables where the limiting distribution is the semicircle distribution. Starting from an estimate for the Kolmogorov distance between the measure of suitably normalized sums of free random variables and the semicircle distribution without any moment condition, we show the free Lindeberg central limit theorem and improve the known results on rates of convergence under the conditions of the existence of the third moments.
The characteristic polynomial of sums of random permutations and regular digraphs
本文研究随机置换矩阵之和 $A_n$ 的归一化特征多项式，证明其收敛于单位圆盘上的随机解析函数，并据此给出随机正则有向图谱隙的初等证明。
We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{á}dl{á}g functions endowed with Skorohod's $J\_1$ topology, to stable distributions in $\mathcal D$. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications, in particular to the empirical process of the renewal-reward process.
We consider a random field, defined on an integer-valued d-dimensional lattice, with covariance function satisfying a condition more general than summability. Such condition appeared in the well-known Newman's conjecture concerning the central limit theorem (CLT) for stationary associated random fields. As was demonstrated by Herrndorf and Shashkin, the conjecture fails already for d=1. In the present paper, we show the validity of modified conjecture leaving intact the mentioned condition on covariance function. Thus we establish, for any positive integer d, a criterion of the CLT validity fo…
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(λ)=\exp{λA_t-λ^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq 0}A_t/{B_t(\log \log B_{…
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